Some Special Dual Direction Curves

Year 2018, , 509 - 517, 30.12.2018

Burak Şahiner

https://doi.org/10.18185/erzifbed.348033

Cited By: 2

Abstract

Bu çalışmada, dual
involüt-evolüt-doğrultu eğrileri, dual Bertrand-doğrultu eğrileri ve dual
Mannheim-doğrultu eğrileri denilen bazı özel bağlantılı eğriler tanımlanmıştır.
Bu bağlantılı dual eğrilerin dual Frenet vektörleri ve dual eğrilikleri
arasında bazı bağıntılar verilmiştir. Ayrıca, dual involüt-evolüt-doğrultu
eğrileri ve dual Mannheim-doğrultu eğrileri kullanarak birim hızlı dual
helislerden birim hızlı dual slant helisler üretmek için kullanışlı yöntemler
sunulmuştur.

Keywords

Bağlantili Eğriler, Doğrultu Eğrileri, Dual Bertrand Eğriler, Dual Helis, Dual Involüt-Evolüt Eğriler, Dual Mannheim Eğriler, Dual Slant Helis

Some Special Dual Direction Curves

Abstract

In this paper, some special
associated dual curves called dual involute-evolute-direction curves, dual
Bertrand-direction curves, and dual Mannheim-direction curves are defined. Some
relations between dual Frenet vectors and dual curvatures of these dual
associated curves are given. Furthermore, useful methods to construct unit
speed dual slant helices from unit speed dual helices by using dual
involute-evolute-direction curves and dual Mannheim-direction curves are
presented.

Keywords

Associated Curves, Direction Curves, Dual Bertrand Curves, Dual Helix, Dual Involute-Evolute Curves, Dual Mannheim Curves, Dual Slant Helix

References

• Arslan Güven, İ., Ağaoğlu, İ. 2014. The properties of Bertrand curves in dual space. International Journal of Physical Sciences, 9(9), 208-213.
• Balgetir, H., Bektaş, M., Inoguchi, J. 2004. Null Bertrand curves in Minkowski 3-space and their characterizations. Note di Matematica, 23(1), 7-13.
• Burke, J.F. 1960. Bertrand curves associated with a pair of curves. Mathematics Magazine, 34(1), 60-62.
• Choi, J.H., Kim, Y.H. 2012. Associated curves of a Frenet curve and their applications. Applied Mathematics and Computation, 218, 9116-9124.
• Choi, J.H., Kim, Y.H., Ali, A.T. 2012. Some associated curves of Frenet non-lightlike curves in . Journal of Mathematical Analysis and Applications, 394, 712-723.
• Güngör, M.A., Tosun, M. 2010. A study on dual Mannheim partner curves. International Mathematical Forum, 5(47), 2319-2330.
• Hacısalihoğlu, H.H. (1983). Hareket Geometrisi ve Kuaterniyonlar Teorisi. Gazi University, Faculty of Arts and Sciences Pub., No: 2, Ankara.
• Izumiya, S., Takeuchi, N. 2002. Generic properties of helices and Bertrand curves. Journal of Geometry, 74, 97-109.
• Kızıltuğ, S. and Önder, M. 2015. Associated curves of Frenet curves in three dimensional compact Lie group. Miskolc Mathematical Notes, 16(2), 953-964.
• Körpınar, T., Sarıaydın, M.T., Turhan, E. 2013. Associated curves according to Bishop frame in Euclidean 3-space. Advanced Modeling and Optimization, 15(3), 713-717.
• Lee, J. W., Choi, J. H., Jin, D. H. 2011. Slant dual Mannheim partner curves in the dual space. International Journal of Contemporary Mathematical Sciences, 6(31), 1535-1544.
• Liu, H., Wang, F. 2008. Mannheim partner curves in 3-space. Journal of Geometry, 88, 120-126.
• Macit, N., Düldül, M. 2014. Some new associated curves of a Frenet curve in and . Turkish Journal of Mathematics, 38, 1023-1037.
• O’Neill, B. 2006. Elementary Differential Geometry. Academic Press.
• Önder, M., Uğurlu, H.H. 2013. Normal and spherical curves in dual space . Mediterr. J. Math., 10, 1527-1537.
• Özkaldı, S., İlarslan, K., Yaylı, Y. 2009. On Mannheim partner curve in dual space. Analele Stiintifice Ale Universitatii Ovidius Constanta, 17(2), 131-142.
• Qian, J., Kim, Y.H. 2015. Directional associated curves of a null curve in Minkowski 3-space. Bulletin of the Korean Mathematical Society, 52(1), 183-200.
• Şenyurt, S., Bilici, M., Çalışkan, M. 2015. Some characterizations for the involute curves in dual space. International Journal of Mathematical Combinatorics, 1, 113-125.
• Turgut, M., Yılmaz, S. 2008. On the Frenet frame and a characterization of space-like Involute-Evolute curve couple in Minkowski space-time. International Mathematical Forum, 3(16), 793-801.
• Veldkamp, G.R. 1976. On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics. Mech. Mach. Theory, 11, 141-156.
• Yang, A.T. (1963). Application of quaternion algebra and dual numbers to the analysis of spatial mechanisms. Doctoral dissertation, Columbia University.
• Yücesan, A., Ayyıldız, N., Çöken, A.C. 2007. On rectifying dual space curves. Rev. Mat. Complut., 20(2), 497-506.